Parabola

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Parabola

In a parabola, all points are equidistant from a point F, called focus and a line called directrix, d. The vertex V is themidpoint between F and d. Then distance from V to F is the same as the distance from V to directrix d. This distance is the absolute value of a numberdenoted p, p then, can be positive or negative, depending whether the parabola opens up or to the right (then p is positive) or if it opens down orto the left (then p is negative).

The vertex is the point we need in the equation, as well as p and the type of parabola: horizontal when theaxis, that is the line through V and F is a horizontal line or vertical if the axis is a vertical line.

The following is a horizontal parabolathat opens to the left, the value of p is negative in this example.

[pic]

In the equation [pic] the vertex is V(-1,2) and 4p= - 12 then p=-3. It is a horizontal parabola that opens to the left, the focus is F(-4, 2), because this point is 3 units to the left (p is negative 3) ofthe vertex. The directrix is x= 1 because this is the line that is 3 units to the right of the vertex.



The following is a vertical parabolawith p positive

[pic]

In the equation [pic] the vertex is V(3,2) and 4p=20 then p=5. It is a vertical parabola that opens up, the focus isF(3, 7), because this point is p=5 units above the vertex, the directrix is y= -3 because this is the line that is p=5 units below the vertex.